Immersed finite element methods are typically confronted by three computational challenges in comparison to standard mesh-fitting finite elements, viz.: i) the numerical evaluation of integrals over cut elements; ii) the imposition of (essential) boundary conditions over the immersed or unfitted boundaries; and iii) the stability of the formulation in relation to small cut elements. A myriad of advanced techniques has been developed over the past decades to resolve these challenges, which has made immersed finite element techniques a competitive simulation strategy for a wide range of problems. This review focuses on the third challenge, i.e., the effects of small cut elements on the performance of immersed finite element methods. We restrict ourselves to a high-level consideration of the first two of these challenges, and we refer the reader to, e.g., Ref. [1]} for a detailed review on these topics.
